The variance of the incipient infinite cluster in two-dimensional percolation

TL;DR

This paper investigates the variance of the size of the incipient infinite cluster in two-dimensional percolation, revealing it scales as n^{91/24+o(1)} under certain conditions, advancing understanding of critical phenomena.

Contribution

It provides a precise asymptotic for the variance of the incipient infinite cluster's size in 2D percolation, which was previously unknown.

Findings

Variance scales as n^{91/24+o(1)}

Confirms the existence of a one-arm path exponent of 5/48

Enhances understanding of critical cluster fluctuations

Abstract

Consider bond percolation on the square lattice. Let ${\cal C}$ be the incipient infinite cluster with the incipient measure $\nu$. If a one-arm path exponent exists and equals $5/48$, it is well known that $E_\nu|{\cal C}\cup [-n,n]^2| = n^{ 91/48+o(1)}$. In this paper, we focus on the variance of $|{\cal C}\cup [-n,n]^2|$ and show that $\sigma^2_\nu (|{\cal C}\cup [-n,n]^2|)= n^{91/24+o(1)}$.